On weak convergence of dyadic scans

I will discuss the limiting behaviour of the scan statistic obtained by calculating likelihood ratio statistics over all contiguous subsets of a sequence of standard Gaussians X1, X2, …, Xn, and the same statistic restricted to subsets with a dyadic number of elements.

In this situation something strange happens: the “complete scan” can be centred and scaled so that it converges weakly to an extreme value distribution but the “dyadic scan” cannot. Instead the maximum fluctuates, similar to what is observed when taking the maximum of a sequence of i.i.d. Geometric random variables. I will give a proof-sketch of this bizarre result. Along the way I will review some powerful tool for the analysis of Gaussian fields which could potentially be useful for other research questions.